Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The unscaled paths of branching Brownian motion

Simon C. Harris and Matthew I. Roberts

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Abstract

For a set AC[0, ∞), we give new results on the growth of the number of particles in a branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. We also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic.

Résumé

Considérons un mouvement Brownien branchant. Nous nous intéressons au nombre de particules dont le chemin reste dans un ensemble fixé AC[0, ∞). Nous montrons qu’il n’est pas nécessaire de renormaliser les chemins. Nous donnons les probabilités de grandes déviations, ainsi qu’une preuve plus sophistiquée pour un résultat concernant la croissance du nombre de particules dans certains ensembles. Nos résultats démontrent que ce nombre de particules peut fortement osciller. Nous obtenons aussi des résultats nouveaux concernant le nombre de particules proches de la frontière du système. Nos méthodes sont purement probabilistes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 579-608.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148211

Digital Object Identifier
doi:10.1214/11-AIHP417

Mathematical Reviews number (MathSciNet)
MR2954267

Zentralblatt MATH identifier
1259.60102

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching Brownian motion Large deviations Survival probability Law of large numbers

Citation

Harris, Simon C.; Roberts, Matthew I. The unscaled paths of branching Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 579--608. doi:10.1214/11-AIHP417. https://projecteuclid.org/euclid.aihp/1334148211


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